As quantumet notes, this derivation has several errors (although it still comes up with an
equation which works under certain special circumstances). Mainly, Bohr assumes (in Step 1) that the
electron is just an ordinary Newtonian object in uniform circular motion. But here, for your viewing
pleasure, is the actual derivation. You'll need to know a fair amount of basic physics and a few
formulas from very basic quantum mechanics.
- F = ke*ze / (r2) = m*v2 / r
- mv2 = k*z*e2 / r
- KE = 0.5mv2 = k*z*e2 / (2*r)
- U = - k*z*e2 / r
- E = KE + U = - k*z*e2 / (2*r)
- L = I*omega (Angular Momentum = Moment of Inertia * Angular Velocity)
- I = mr2
- w = v / r
- L = (mr2)*(v / r) = mvr
- mvr = n*h/(2*pi)
- v = nh / (2*pi*m*r)
- m*(nh / 2*pi*m*r)2 = k*z*e2 / r
- r = (n2 / z)*(h2epsilon0 / (pi*m*e2))
- E = -kze2 / (2*r) = -kze2 / (2*(n2 /
z)*(h2epsilon0 / (pi*m*e2)))
- E = -(z / n)2 * ((e4m) / (8*(epsilon0h)2))
- E = (R/h)*(1/n)2
- h*c/lambda = (R/h)*(1 / nf2 - 1 / ni2)
- 1 / lambda = R*(1 / nf2 - 1 / ni2)
Notes:
- Step 1: For a single electron orbiting around a nucleus, where e is the basic electrical
charge, k is a constant, r is the radius, z is the number of protons, and m is the mass of an electron.
- Step 10: This was a big leap of faith for Bohr. As de Broglie later proved in 1923, this
step works because the electron moves as a standing particle wave around the nucleus. For more
information, see de Broglie's explanation of Bohr's assumption about angular momentum
- Step 12: Plugging in the results for v to step 2
- Step 13: Since k = 1 / (4*pi*epsilon0)
- Step 14: Plugging in the results for r to step 6
- Step 16: Assuming that this is a hydrogen atom, and therefore z = 1. Also, the whole mess of constants out in front simplifies down to Rydberg's constant, R (actually R divided by h, but the h kindly disappears in the next step).
For further reference:
Cutnell, John D. and Kenneth W. Johnson, Physics. New York: John Wiley & Sons, 1998 (4th
edition),
or any good physics book.